Kompany took charge of Anderlecht in August 2019, starting out as player-manager, before deciding to concentrate on playing after the Belgian club recorded their worst start to a league season.
In a statement on Anderlecht's website, Kompany said: "Today only one feeling prevails: I am proud that I was able to start this new chapter at the club of my heart. I have now been a player and a coach of RSC Anderlecht, but above all I remain a loyal fan."
He returned to the dugout a year later and in his first season, he guided Anderlecht to fourth in the Belgian first division, and finished third, 13 points behind regular season leaders Union Saint-Gilloise, in 2021/22.
The 36-year-old is believed to be keen on the challenge despite the Clarets' relegation from the Premier League, The Guardian reports.
It's understood Kompany is on a three-man shortlist of candidates to succeed Sean Dyche, who was sacked in April, at Turf Moor but no decision has been over the role as yet.
Kompany's wife, Carla, is from Manchester meaning a return to the north west of England is likely to play a factor in his decision.
Mike Jackson has been in temporary charge ever since, and managed to secure 11 points from eight games. However, it was not enough as the Clarets suffered relegation on the final day of the Premier League season, following a 2-1 defeat at home to Newcastle.
Jackson has remained tight-lipped on whether he wants to continue in his caretaker role full-time when he spoke to reporters last week ahead of that final-day defeat to the Magpies which confirmed their relegation.
"I've just not thought about [the job] at all. All my thoughts have been how can we achieve what we want to do", he said.
"We've not spoken about it. We've not had time to even think about that. My full focus has been on trying to achieve this.
"Let's see what happens on Monday morning. Fingers crossed everything goes how we want, then we'll have a good Monday."